Question: Derek sent a chain letter to his friends, asking them to forward the letter to more friends. The group of people who receive the email gains $\dfrac{9}{10}$ of its size every $3$ weeks, and can be modeled by a function, $P$, which depends on the amount of time, $t$ (in weeks). Derek initially sent the chain letter to $40$ friends. Write a function that models the group of people who receive the email $t$ weeks since Derek initially sent the chain letter. $P(t) = $
Explanation: The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial number of people who receive the email is $40$, and the group of people who receive the email gains $\dfrac{9}{10}$ of its size every $3$ weeks. Note that gaining $\dfrac{9}{10}$ is the same as being multiplied by $\dfrac{19}{10}$. [Why?] This means that the initial quantity is $A=40$ and the factor is $B=\dfrac{19}{10}$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $\dfrac{19}{10}$ every $3$ weeks. Finding the expression in the exponent We know that the number of people who receive the email is multiplied by $\dfrac{19}{10}$ every $3$ weeks. This means that each time $t$ increases by $3$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{3}$. When Derek initially sent the e-mail, the number of people who have received the e-mail has not begun to increase yet. So $P(0) = 40$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{3}$. Summary We found that the following function models the number of people who receive the email $t$ weeks since Derek initially sent the chain letter. P ( t ) = 40 ⋅ ( 19 10 ) t 3 P(t)=40\cdot \left(\dfrac{19}{10}\right)\^{ \frac{t}{3}}